Partially Observable Markov Decision Processes. Mausam (slides by Dieter Fox)

Transcription

1 Partially Observable Markov Decision Processes Mausam (slides by Dieter Fox)

2 Stochastic Planning: MDPs Static Environment Fully Observable Perfect What action next? Stochastic Instantaneous Percepts Actions 2

3 Partially Observable MDPs Static Environment Partially Observable Noisy What action next? Stochastic Instantaneous Percepts Actions 3

4 Stochastic, Fully Observable 4

5 Stochastic, Partially Observable 5

6 POMDPs In POMDPs we apply the very same idea as in MDPs. Since the state is not observable, the agent has to make its decisions based on the belief state which is a posterior distribution over states. Let b be the belief of the agent about the current state POMDPs compute a value function over belief space: a a γ b, a 6

7 POMDPs Each belief is a probability distribution, value fn is a function of an entire probability distribution. Problematic, since probability distributions are continuous. Also, we have to deal with huge complexity of belief spaces. For finite worlds with finite state, action, and observation spaces and finite horizons, we can represent the value functions by piecewise linear functions. 7

8 Applications Robotic control helicopter maneuvering, autonomous vehicles Mars rover - path planning, oversubscription planning elevator planning Game playing - backgammon, tetris, checkers Neuroscience Computational Finance, Sequential Auctions Assisting elderly in simple tasks Spoken dialog management Communication Networks switching, routing, flow control War planning, evacuation planning 8

9 An Illustrative Example measurements state x action u 3 state x 2 measurements z z u x 3 u u 3 u x 2 u u z z 2 actions u, u 2 payoff payoff 9

10 The Parameters of the Example The actions u and u 2 are terminal actions. The action u 3 is a sensing action that potentially leads to a state transition. The horizon is finite and =. 0

11 Payoff in POMDPs In MDPs, the payoff (or return) depended on the state of the system. In POMDPs, however, the true state is not exactly known. Therefore, we compute the expected payoff by integrating over all states:

12 Payoffs in Our Example () If we are totally certain that we are in state x and execute action u, we receive a reward of -00 If, on the other hand, we definitely know that we are in x 2 and execute u, the reward is +00. In between it is the linear combination of the extreme values weighted by the probabilities 2

13 Payoffs in Our Example (2) 3

14 The Resulting Policy for T= Given we have a finite POMDP with T=, we would use V (b) to determine the optimal policy. In our example, the optimal policy for T= is This is the upper thick graph in the diagram. 4

15 Piecewise Linearity, Convexity The resulting value function V (b) is the maximum of the three functions at each point It is piecewise linear and convex. 5

16 Pruning If we carefully consider V (b), we see that only the first two components contribute. The third component can therefore safely be pruned away from V (b). 6

17 Increasing the Time Horizon Assume the robot can make an observation before deciding on an action. V (b) 7

18 Increasing the Time Horizon Assume the robot can make an observation before deciding on an action. Suppose the robot perceives z for which p(z x )=0.7 and p(z x 2 )=0.3. Given the observation z we update the belief using Bayes rule. p' p' 2 p( z 0.7 p p( z ) 0.3( p p( z ) ) 0.7 p ) 0.3( p ) 0.4 p 0.3 8

19 Value Function V (b) b (b z ) V (b z ) 9

20 Increasing the Time Horizon Assume the robot can make an observation before deciding on an action. Suppose the robot perceives z for which p(z x )=0.7 and p(z x 2 )=0.3. Given the observation z we update the belief using Bayes rule. Thus V (b z ) is given by 20

21 Expected Value after Measuring Since we do not know in advance what the next measurement will be, we have to compute the expected belief ) ( ) ( ) ( ) ( ) ( ) ( )] ( [ ) ( i i i i i i i i i z p x z p V z p p x z p V z p z b V z p z b V E b V 2

22 Expected Value after Measuring Since we do not know in advance what the next measurement will be, we have to compute the expected belief 22

23 Resulting Value Function The four possible combinations yield the following function which then can be simplified and pruned. 23

24 Value Function p(z ) V (b z ) b (b z ) \bar{v} (b) p(z 2 ) V 2 (b z 2 ) 24

25 State Transitions (Prediction) When the agent selects u 3 its state potentially changes. When computing the value function, we have to take these potential state changes into account. 25

26 State Transitions (Prediction) When the agent selects u 3 its state potentially changes. When computing the value function, we have to take these potential state changes into account. 26

27 Resulting Value Function after executing u 3 Taking the state transitions into account, we finally obtain. 27

28 Value Function after executing u 3 \bar{v} (b) \bar{v} (b u 3 ) 28

29 Value Function for T=2 Taking into account that the agent can either directly perform u or u 2 or first u 3 and then u or u 2, we obtain (after pruning) 29

30 Graphical Representation of V 2 (b) u optimal u 2 optimal unclear outcome of measuring is important here 30

31 Deep Horizons and Pruning We have now completed a full backup in belief space. This process can be applied recursively. The value functions for T=0 and T=20 are 3

32 Deep Horizons and Pruning 32

33 33

34 Why Pruning is Essential Each update introduces additional linear components to V. Each measurement squares the number of linear components. Thus, an unpruned value function for T=20 includes more than 0 547,864 linear functions. At T=30 we have 0 56,02,337 linear functions. The pruned value functions at T=20, in comparison, contains only 2 linear components. The combinatorial explosion of linear components in the value function are the major reason why POMDPs are impractical for most applications. 34

35 POMDP Summary POMDPs compute the optimal action in partially observable, stochastic domains. For finite horizon problems, the resulting value functions are piecewise linear and convex. In each iteration the number of linear constraints grows exponentially. POMDPs so far have only been applied successfully to very small state spaces with small numbers of possible observations and actions. 35